Elliptic orbit

In astrodynamics or celestial mechanics a elliptic orbit is an orbit with the eccentricity greater than 0 and less than 1.

Specific energy of an elliptical orbit is negative.

Contents

1 Velocity

Velocity

Under standard assumptions the orbital velocity ( $v\,$ ) of a body traveling along elliptic orbit can be computed as:

$v=\sqrt{2\mu\left({1\over{r}}-{1\over{2a}}\right)}$

where:

Conclusion:

Velocity does not depend on eccentricity but is determined by length of semi-major axis ( $a\,\!$ ),
Velocity equation is similar to that for hyperbolic trajectory with the difference that for the latter one ${1\over{2a}}$ is positive.

Under standard assumptions the orbital period ( $T\,\!$ ) of a body traveling along elliptic orbit can be computed as:

$T={2\pi\over{\sqrt{\mu}}}a^{3\over{2}}$

where:

Conclusions:

The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis ( $a\,\!$ ),
The orbital period does not depend on the eccentricity (See also: Kepler's third law).

Under standard assumptions, specific orbital energy ( $\epsilon\,$ ) of elliptic orbit is negative and the orbital energy conservation equation for this orbit takes form:

${v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0$

where:

Conclusions:

Specific energy for elliptic orbits is independent of eccentricity and is determined only by semi-major axis of the ellipse.

Using the virial theorem we find:

the time-average of the specific potential energy is equal to 2ε
- the time-average of r^-1 is a^-1
the time-average of the specific kinetic energy is equal to -ε

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